7.4. GLM Fragility Functions

postprocessor.calculate_glm_fragility(imls, edps, damage_thresholds, intensities=np.round(np.geomspace(0.05, 10.0, 50), 3), fragility_method='logit')[source]

Computes non-parametric fragility functions using Generalized Linear Models (GLM) with either a Logit or Probit link function.

Parameters:

  • imls (array-like) – Intensity Measure Levels (IMLs) corresponding to each observation.

  • edps (array-like) – Engineering Demand Parameters (EDPs) representing structural response values.

  • damage_thresholds (array-like) – List of thresholds defining different damage states.

  • intensities (array-like, optional) – Intensity measure values at which probabilities of exceedance (PoEs) are evaluated. Defaults to np.round(np.geomspace(0.05, 10.0, 50), 3).

  • fragility_method (str, optional) –

    Specifies the GLM model to be used for fragility function fitting. Options:

    • 'logit' (default): Uses a logistic regression model.

    • 'probit': Uses a probit regression model.

Returns:

poes – A 2D array where each column represents the probability of exceeding a specific damage state at each intensity level.

Return type:

ndarray

References

1) Charvet, I., Ioannou, I., Rossetto, T., Suppasri, A., and Imamura, F.: Empirical fragility assessment of buildings affected by the 2011 Great East Japan tsunami using improved statistical models, Nat. Hazards, 73, 951-973, 2014.

2) Lahcene, E., Ioannou, I., Suppasri, A., Pakoksung, K., Paulik, R., Syamsidik, S., Bouchette, F., and Imamura, F.: Characteristics of building fragility curves for seismic and non-seismic tsunamis: case studies of the 2018 Sunda Strait, 2018 Sulawesi-Palu, and 2004 Indian Ocean tsunamis, Nat. Hazards Earth Syst. Sci., 21, 2313-2344, https://doi.org/10.5194/nhess-21-2313-2021, 2021.

3) Lallemant, D., Kiremidjian, A., and Burton, H. (2015), Statistical procedures for developing earthquake damage fragility curves. Earthquake Engng Struct. Dyn., 44, 1373-1389. doi: 10.1002/eqe.2522.

4) Jalayer, F., Ebrahamian, H., Trevlopoulos, K., and Bradley, B. (2023). Empirical tsunami fragility modelling for hierarchical damage levels. Natural Hazards and Earth System Sciences, 23(2), 909-931. https://doi.org/10.5194/nhess-23-909-2023

Theoretical Background

Generalised Linear Model (GLM) fragility fitting uses binary regression to estimate exceedance probabilities directly from damage observations, without assuming a lognormal form (Lallemant et al., 2015).

Binary response model

Let \(y_j \in \{0, 1\}\) indicate whether record \(j\) caused exceedance of the damage-state threshold. The exceedance probability is modelled as:

\[P(\text{DS} \geq ds_i \mid \text{IM}_j) = g^{-1}(\eta_j), \quad \eta_j = \beta_0 + \beta_1 \ln(\text{IM}_j)\]

where \(g^{-1}\) is the inverse link function. Two link functions are available:

  • Probit: \(g^{-1}(\eta) = \Phi(\eta)\) (standard normal CDF). This recovers the lognormal fragility model when the predictor is \(\ln(\text{IM})\).

  • Logit: \(g^{-1}(\eta) = 1/(1 + e^{-\eta})\) (logistic sigmoid).

Maximum likelihood estimation

Parameters \((\beta_0, \beta_1)\) are estimated by maximising the Bernoulli log-likelihood:

\[\mathcal{L}(\beta_0, \beta_1) = \sum_{j=1}^{N} \Bigl[ y_j \ln p_j + (1 - y_j) \ln(1 - p_j) \Bigr], \quad p_j = g^{-1}(\beta_0 + \beta_1 \ln \text{IM}_j)\]

Dispersion

The record-to-record dispersion is derived from the fitted coefficients. For the probit link, \(\beta_{\text{r2r}} = 1/\beta_1\). Total dispersion is then:

\[\beta_{\text{total}} = \sqrt{\beta_{\text{r2r}}^2 + \beta_{\text{b2b}}^2 + \beta_{\text{DS}}^2}\]